And on that note about the Central Line, with all that clay around it, it's no wonder the line was colored red.The NYC Subway lines are all what we'd call in London "Subsurface lines". They run just below street level, in fairly large tunnels. Some of London's lines are like this too, and we have AC on those.
But about 2/3 of the Tube network is what are called "deep tube" lines. These are much deeper below the ground (up to 50 meters in places) and in narrow, bored tunnels that leave very little clearance between the trains and the tunnel walls. It's on these lines that air conditioning is very difficult to install.

Why is London's Central line so hot? Science has the answer
The London Underground is hot. But nowhere is hotter than the Central line, which is routinely so hot that it exceeds the EU limit at which it is legal to transport cows, sheep and pigs